Let $\dot{X} = F(X)$ be a non-linear dynamical system. I'm interested in knowing if there are any existence theorems for an equilibrium point, that is, an $X^*$ that satisfies: $$ F(X^*) = 0 $$
I know that in the linear case $(F(X) = AX)$, it's trivial to establish that $X^* = 0$ is the equilibrium point; but I haven't found any theorem that guarantees the existence of equilibrium points in the non-linear case.
In case there is a theorem that guarantees existence, I'm also interested in a theorem that could tell us something about the uniqueness (or lack thereof) of equilibrium points.
So far, I've only come up with the idea that one should check whether $F(X) + X = G(X)$ satisfies Brouwer's fixed point theorem to check if the system has an equilibrium point, but I don't know if this is the right approach.
Thanks!